\(\int \frac {260 x+62 x^2+2 x^3+(e^x (125 x+30 x^2+x^3)+e^x (125+30 x+x^2) \log (25+x)) \log (x^2+2 x \log (25+x)+\log ^2(25+x))+(125 x+30 x^2+x^3+(125+30 x+x^2) \log (25+x)) \log (x^2+2 x \log (25+x)+\log ^2(25+x)) \log (\log (x^2+2 x \log (25+x)+\log ^2(25+x)))+((e^x (-25 x-x^2)+e^x (-25-x) \log (25+x)) \log (x^2+2 x \log (25+x)+\log ^2(25+x))+(-25 x^2-x^3+(-25 x-x^2) \log (25+x)) \log (x^2+2 x \log (25+x)+\log ^2(25+x)) \log (\log (x^2+2 x \log (25+x)+\log ^2(25+x)))) \log (e^x+x \log (\log (x^2+2 x \log (25+x)+\log ^2(25+x))))}{(e^x (625 x+275 x^2+35 x^3+x^4)+e^x (625+275 x+35 x^2+x^3) \log (25+x)) \log (x^2+2 x \log (25+x)+\log ^2(25+x))+(625 x^2+275 x^3+35 x^4+x^5+(625 x+275 x^2+35 x^3+x^4) \log (25+x)) \log (x^2+2 x \log (25+x)+\log ^2(25+x)) \log (\log (x^2+2 x \log (25+x)+\log ^2(25+x)))} \, dx\) [283]

Optimal result

Integrand size = 411, antiderivative size = 23 \[ \int \frac {260 x+62 x^2+2 x^3+\left (e^x \left (125 x+30 x^2+x^3\right )+e^x \left (125+30 x+x^2\right ) \log (25+x)\right ) \log \left (x^2+2 x \log (25+x)+\log ^2(25+x)\right )+\left (125 x+30 x^2+x^3+\left (125+30 x+x^2\right ) \log (25+x)\right ) \log \left (x^2+2 x \log (25+x)+\log ^2(25+x)\right ) \log \left (\log \left (x^2+2 x \log (25+x)+\log ^2(25+x)\right )\right )+\left (\left (e^x \left (-25 x-x^2\right )+e^x (-25-x) \log (25+x)\right ) \log \left (x^2+2 x \log (25+x)+\log ^2(25+x)\right )+\left (-25 x^2-x^3+\left (-25 x-x^2\right ) \log (25+x)\right ) \log \left (x^2+2 x \log (25+x)+\log ^2(25+x)\right ) \log \left (\log \left (x^2+2 x \log (25+x)+\log ^2(25+x)\right )\right )\right ) \log \left (e^x+x \log \left (\log \left (x^2+2 x \log (25+x)+\log ^2(25+x)\right )\right )\right )}{\left (e^x \left (625 x+275 x^2+35 x^3+x^4\right )+e^x \left (625+275 x+35 x^2+x^3\right ) \log (25+x)\right ) \log \left (x^2+2 x \log (25+x)+\log ^2(25+x)\right )+\left (625 x^2+275 x^3+35 x^4+x^5+\left (625 x+275 x^2+35 x^3+x^4\right ) \log (25+x)\right ) \log \left (x^2+2 x \log (25+x)+\log ^2(25+x)\right ) \log \left (\log \left (x^2+2 x \log (25+x)+\log ^2(25+x)\right )\right )} \, dx=\frac {\log \left (e^x+x \log \left (\log \left ((x+\log (25+x))^2\right )\right )\right )}{5+x} \]

[Out]

ln(x*ln(ln((ln(x+25)+x)^2))+exp(x))/(5+x)

Rubi [F]

\[ \int \frac {260 x+62 x^2+2 x^3+\left (e^x \left (125 x+30 x^2+x^3\right )+e^x \left (125+30 x+x^2\right ) \log (25+x)\right ) \log \left (x^2+2 x \log (25+x)+\log ^2(25+x)\right )+\left (125 x+30 x^2+x^3+\left (125+30 x+x^2\right ) \log (25+x)\right ) \log \left (x^2+2 x \log (25+x)+\log ^2(25+x)\right ) \log \left (\log \left (x^2+2 x \log (25+x)+\log ^2(25+x)\right )\right )+\left (\left (e^x \left (-25 x-x^2\right )+e^x (-25-x) \log (25+x)\right ) \log \left (x^2+2 x \log (25+x)+\log ^2(25+x)\right )+\left (-25 x^2-x^3+\left (-25 x-x^2\right ) \log (25+x)\right ) \log \left (x^2+2 x \log (25+x)+\log ^2(25+x)\right ) \log \left (\log \left (x^2+2 x \log (25+x)+\log ^2(25+x)\right )\right )\right ) \log \left (e^x+x \log \left (\log \left (x^2+2 x \log (25+x)+\log ^2(25+x)\right )\right )\right )}{\left (e^x \left (625 x+275 x^2+35 x^3+x^4\right )+e^x \left (625+275 x+35 x^2+x^3\right ) \log (25+x)\right ) \log \left (x^2+2 x \log (25+x)+\log ^2(25+x)\right )+\left (625 x^2+275 x^3+35 x^4+x^5+\left (625 x+275 x^2+35 x^3+x^4\right ) \log (25+x)\right ) \log \left (x^2+2 x \log (25+x)+\log ^2(25+x)\right ) \log \left (\log \left (x^2+2 x \log (25+x)+\log ^2(25+x)\right )\right )} \, dx=\int \frac {260 x+62 x^2+2 x^3+\left (e^x \left (125 x+30 x^2+x^3\right )+e^x \left (125+30 x+x^2\right ) \log (25+x)\right ) \log \left (x^2+2 x \log (25+x)+\log ^2(25+x)\right )+\left (125 x+30 x^2+x^3+\left (125+30 x+x^2\right ) \log (25+x)\right ) \log \left (x^2+2 x \log (25+x)+\log ^2(25+x)\right ) \log \left (\log \left (x^2+2 x \log (25+x)+\log ^2(25+x)\right )\right )+\left (\left (e^x \left (-25 x-x^2\right )+e^x (-25-x) \log (25+x)\right ) \log \left (x^2+2 x \log (25+x)+\log ^2(25+x)\right )+\left (-25 x^2-x^3+\left (-25 x-x^2\right ) \log (25+x)\right ) \log \left (x^2+2 x \log (25+x)+\log ^2(25+x)\right ) \log \left (\log \left (x^2+2 x \log (25+x)+\log ^2(25+x)\right )\right )\right ) \log \left (e^x+x \log \left (\log \left (x^2+2 x \log (25+x)+\log ^2(25+x)\right )\right )\right )}{\left (e^x \left (625 x+275 x^2+35 x^3+x^4\right )+e^x \left (625+275 x+35 x^2+x^3\right ) \log (25+x)\right ) \log \left (x^2+2 x \log (25+x)+\log ^2(25+x)\right )+\left (625 x^2+275 x^3+35 x^4+x^5+\left (625 x+275 x^2+35 x^3+x^4\right ) \log (25+x)\right ) \log \left (x^2+2 x \log (25+x)+\log ^2(25+x)\right ) \log \left (\log \left (x^2+2 x \log (25+x)+\log ^2(25+x)\right )\right )} \, dx \]

[In]

Int[(260*x + 62*x^2 + 2*x^3 + (E^x*(125*x + 30*x^2 + x^3) + E^x*(125 + 30*x + x^2)*Log[25 + x])*Log[x^2 + 2*x*
Log[25 + x] + Log[25 + x]^2] + (125*x + 30*x^2 + x^3 + (125 + 30*x + x^2)*Log[25 + x])*Log[x^2 + 2*x*Log[25 +
x] + Log[25 + x]^2]*Log[Log[x^2 + 2*x*Log[25 + x] + Log[25 + x]^2]] + ((E^x*(-25*x - x^2) + E^x*(-25 - x)*Log[
25 + x])*Log[x^2 + 2*x*Log[25 + x] + Log[25 + x]^2] + (-25*x^2 - x^3 + (-25*x - x^2)*Log[25 + x])*Log[x^2 + 2*
x*Log[25 + x] + Log[25 + x]^2]*Log[Log[x^2 + 2*x*Log[25 + x] + Log[25 + x]^2]])*Log[E^x + x*Log[Log[x^2 + 2*x*
Log[25 + x] + Log[25 + x]^2]]])/((E^x*(625*x + 275*x^2 + 35*x^3 + x^4) + E^x*(625 + 275*x + 35*x^2 + x^3)*Log[
25 + x])*Log[x^2 + 2*x*Log[25 + x] + Log[25 + x]^2] + (625*x^2 + 275*x^3 + 35*x^4 + x^5 + (625*x + 275*x^2 + 3
5*x^3 + x^4)*Log[25 + x])*Log[x^2 + 2*x*Log[25 + x] + Log[25 + x]^2]*Log[Log[x^2 + 2*x*Log[25 + x] + Log[25 +
x]^2]]),x]

[Out]

Log[5 + x] + 2*Defer[Int][1/((x + Log[25 + x])*Log[(x + Log[25 + x])^2]*(E^x + x*Log[Log[(x + Log[25 + x])^2]]
)), x] - (21*Defer[Int][1/((5 + x)*(x + Log[25 + x])*Log[(x + Log[25 + x])^2]*(E^x + x*Log[Log[(x + Log[25 + x
])^2]])), x])/2 + (5*Defer[Int][1/((25 + x)*(x + Log[25 + x])*Log[(x + Log[25 + x])^2]*(E^x + x*Log[Log[(x + L
og[25 + x])^2]])), x])/2 + 6*Defer[Int][Log[Log[(x + Log[25 + x])^2]]/((x + Log[25 + x])*(E^x + x*Log[Log[(x +
Log[25 + x])^2]])), x] - Defer[Int][(x*Log[Log[(x + Log[25 + x])^2]])/((x + Log[25 + x])*(E^x + x*Log[Log[(x
+ Log[25 + x])^2]])), x] - 30*Defer[Int][Log[Log[(x + Log[25 + x])^2]]/((5 + x)*(x + Log[25 + x])*(E^x + x*Log
[Log[(x + Log[25 + x])^2]])), x] - Defer[Int][(Log[25 + x]*Log[Log[(x + Log[25 + x])^2]])/((x + Log[25 + x])*(
E^x + x*Log[Log[(x + Log[25 + x])^2]])), x] + 6*Defer[Int][(Log[25 + x]*Log[Log[(x + Log[25 + x])^2]])/((5 + x
)*(x + Log[25 + x])*(E^x + x*Log[Log[(x + Log[25 + x])^2]])), x] - Defer[Int][Log[E^x + x*Log[Log[(x + Log[25
+ x])^2]]]/(5 + x)^2, x]

Rubi steps \begin{align*} \text {integral}& = \int \frac {260 x+62 x^2+2 x^3+e^x \left (125+30 x+x^2\right ) (x+\log (25+x)) \log \left ((x+\log (25+x))^2\right )+\left (125+30 x+x^2\right ) (x+\log (25+x)) \log \left ((x+\log (25+x))^2\right ) \log \left (\log \left ((x+\log (25+x))^2\right )\right )-(25+x) (x+\log (25+x)) \log \left ((x+\log (25+x))^2\right ) \left (e^x+x \log \left (\log \left ((x+\log (25+x))^2\right )\right )\right ) \log \left (e^x+x \log \left (\log \left ((x+\log (25+x))^2\right )\right )\right )}{(5+x)^2 (25+x) (x+\log (25+x)) \log \left ((x+\log (25+x))^2\right ) \left (e^x+x \log \left (\log \left ((x+\log (25+x))^2\right )\right )\right )} \, dx \\ & = \int \left (-\frac {-52 x-2 x^2-25 x \log \left ((x+\log (25+x))^2\right ) \log \left (\log \left ((x+\log (25+x))^2\right )\right )+24 x^2 \log \left ((x+\log (25+x))^2\right ) \log \left (\log \left ((x+\log (25+x))^2\right )\right )+x^3 \log \left ((x+\log (25+x))^2\right ) \log \left (\log \left ((x+\log (25+x))^2\right )\right )-25 \log (25+x) \log \left ((x+\log (25+x))^2\right ) \log \left (\log \left ((x+\log (25+x))^2\right )\right )+24 x \log (25+x) \log \left ((x+\log (25+x))^2\right ) \log \left (\log \left ((x+\log (25+x))^2\right )\right )+x^2 \log (25+x) \log \left ((x+\log (25+x))^2\right ) \log \left (\log \left ((x+\log (25+x))^2\right )\right )}{(5+x) (25+x) (x+\log (25+x)) \log \left ((x+\log (25+x))^2\right ) \left (e^x+x \log \left (\log \left ((x+\log (25+x))^2\right )\right )\right )}+\frac {5+x-\log \left (e^x+x \log \left (\log \left ((x+\log (25+x))^2\right )\right )\right )}{(5+x)^2}\right ) \, dx \\ & = -\int \frac {-52 x-2 x^2-25 x \log \left ((x+\log (25+x))^2\right ) \log \left (\log \left ((x+\log (25+x))^2\right )\right )+24 x^2 \log \left ((x+\log (25+x))^2\right ) \log \left (\log \left ((x+\log (25+x))^2\right )\right )+x^3 \log \left ((x+\log (25+x))^2\right ) \log \left (\log \left ((x+\log (25+x))^2\right )\right )-25 \log (25+x) \log \left ((x+\log (25+x))^2\right ) \log \left (\log \left ((x+\log (25+x))^2\right )\right )+24 x \log (25+x) \log \left ((x+\log (25+x))^2\right ) \log \left (\log \left ((x+\log (25+x))^2\right )\right )+x^2 \log (25+x) \log \left ((x+\log (25+x))^2\right ) \log \left (\log \left ((x+\log (25+x))^2\right )\right )}{(5+x) (25+x) (x+\log (25+x)) \log \left ((x+\log (25+x))^2\right ) \left (e^x+x \log \left (\log \left ((x+\log (25+x))^2\right )\right )\right )} \, dx+\int \frac {5+x-\log \left (e^x+x \log \left (\log \left ((x+\log (25+x))^2\right )\right )\right )}{(5+x)^2} \, dx \\ & = -\int \frac {-2 x (26+x)+\left (-25+24 x+x^2\right ) (x+\log (25+x)) \log \left ((x+\log (25+x))^2\right ) \log \left (\log \left ((x+\log (25+x))^2\right )\right )}{(5+x) (25+x) (x+\log (25+x)) \log \left ((x+\log (25+x))^2\right ) \left (e^x+x \log \left (\log \left ((x+\log (25+x))^2\right )\right )\right )} \, dx+\int \left (\frac {1}{5+x}-\frac {\log \left (e^x+x \log \left (\log \left ((x+\log (25+x))^2\right )\right )\right )}{(5+x)^2}\right ) \, dx \\ & = \log (5+x)-\int \left (\frac {-52 x-2 x^2-25 x \log \left ((x+\log (25+x))^2\right ) \log \left (\log \left ((x+\log (25+x))^2\right )\right )+24 x^2 \log \left ((x+\log (25+x))^2\right ) \log \left (\log \left ((x+\log (25+x))^2\right )\right )+x^3 \log \left ((x+\log (25+x))^2\right ) \log \left (\log \left ((x+\log (25+x))^2\right )\right )-25 \log (25+x) \log \left ((x+\log (25+x))^2\right ) \log \left (\log \left ((x+\log (25+x))^2\right )\right )+24 x \log (25+x) \log \left ((x+\log (25+x))^2\right ) \log \left (\log \left ((x+\log (25+x))^2\right )\right )+x^2 \log (25+x) \log \left ((x+\log (25+x))^2\right ) \log \left (\log \left ((x+\log (25+x))^2\right )\right )}{20 (5+x) (x+\log (25+x)) \log \left ((x+\log (25+x))^2\right ) \left (e^x+x \log \left (\log \left ((x+\log (25+x))^2\right )\right )\right )}-\frac {-52 x-2 x^2-25 x \log \left ((x+\log (25+x))^2\right ) \log \left (\log \left ((x+\log (25+x))^2\right )\right )+24 x^2 \log \left ((x+\log (25+x))^2\right ) \log \left (\log \left ((x+\log (25+x))^2\right )\right )+x^3 \log \left ((x+\log (25+x))^2\right ) \log \left (\log \left ((x+\log (25+x))^2\right )\right )-25 \log (25+x) \log \left ((x+\log (25+x))^2\right ) \log \left (\log \left ((x+\log (25+x))^2\right )\right )+24 x \log (25+x) \log \left ((x+\log (25+x))^2\right ) \log \left (\log \left ((x+\log (25+x))^2\right )\right )+x^2 \log (25+x) \log \left ((x+\log (25+x))^2\right ) \log \left (\log \left ((x+\log (25+x))^2\right )\right )}{20 (25+x) (x+\log (25+x)) \log \left ((x+\log (25+x))^2\right ) \left (e^x+x \log \left (\log \left ((x+\log (25+x))^2\right )\right )\right )}\right ) \, dx-\int \frac {\log \left (e^x+x \log \left (\log \left ((x+\log (25+x))^2\right )\right )\right )}{(5+x)^2} \, dx \\ & = \log (5+x)-\frac {1}{20} \int \frac {-52 x-2 x^2-25 x \log \left ((x+\log (25+x))^2\right ) \log \left (\log \left ((x+\log (25+x))^2\right )\right )+24 x^2 \log \left ((x+\log (25+x))^2\right ) \log \left (\log \left ((x+\log (25+x))^2\right )\right )+x^3 \log \left ((x+\log (25+x))^2\right ) \log \left (\log \left ((x+\log (25+x))^2\right )\right )-25 \log (25+x) \log \left ((x+\log (25+x))^2\right ) \log \left (\log \left ((x+\log (25+x))^2\right )\right )+24 x \log (25+x) \log \left ((x+\log (25+x))^2\right ) \log \left (\log \left ((x+\log (25+x))^2\right )\right )+x^2 \log (25+x) \log \left ((x+\log (25+x))^2\right ) \log \left (\log \left ((x+\log (25+x))^2\right )\right )}{(5+x) (x+\log (25+x)) \log \left ((x+\log (25+x))^2\right ) \left (e^x+x \log \left (\log \left ((x+\log (25+x))^2\right )\right )\right )} \, dx+\frac {1}{20} \int \frac {-52 x-2 x^2-25 x \log \left ((x+\log (25+x))^2\right ) \log \left (\log \left ((x+\log (25+x))^2\right )\right )+24 x^2 \log \left ((x+\log (25+x))^2\right ) \log \left (\log \left ((x+\log (25+x))^2\right )\right )+x^3 \log \left ((x+\log (25+x))^2\right ) \log \left (\log \left ((x+\log (25+x))^2\right )\right )-25 \log (25+x) \log \left ((x+\log (25+x))^2\right ) \log \left (\log \left ((x+\log (25+x))^2\right )\right )+24 x \log (25+x) \log \left ((x+\log (25+x))^2\right ) \log \left (\log \left ((x+\log (25+x))^2\right )\right )+x^2 \log (25+x) \log \left ((x+\log (25+x))^2\right ) \log \left (\log \left ((x+\log (25+x))^2\right )\right )}{(25+x) (x+\log (25+x)) \log \left ((x+\log (25+x))^2\right ) \left (e^x+x \log \left (\log \left ((x+\log (25+x))^2\right )\right )\right )} \, dx-\int \frac {\log \left (e^x+x \log \left (\log \left ((x+\log (25+x))^2\right )\right )\right )}{(5+x)^2} \, dx \\ & = \log (5+x)-\frac {1}{20} \int \frac {-2 x (26+x)+\left (-25+24 x+x^2\right ) (x+\log (25+x)) \log \left ((x+\log (25+x))^2\right ) \log \left (\log \left ((x+\log (25+x))^2\right )\right )}{(5+x) (x+\log (25+x)) \log \left ((x+\log (25+x))^2\right ) \left (e^x+x \log \left (\log \left ((x+\log (25+x))^2\right )\right )\right )} \, dx+\frac {1}{20} \int \frac {-2 x (26+x)+\left (-25+24 x+x^2\right ) (x+\log (25+x)) \log \left ((x+\log (25+x))^2\right ) \log \left (\log \left ((x+\log (25+x))^2\right )\right )}{(25+x) (x+\log (25+x)) \log \left ((x+\log (25+x))^2\right ) \left (e^x+x \log \left (\log \left ((x+\log (25+x))^2\right )\right )\right )} \, dx-\int \frac {\log \left (e^x+x \log \left (\log \left ((x+\log (25+x))^2\right )\right )\right )}{(5+x)^2} \, dx \\ & = \log (5+x)-\frac {1}{20} \int \left (-\frac {52 x}{(5+x) (x+\log (25+x)) \log \left ((x+\log (25+x))^2\right ) \left (e^x+x \log \left (\log \left ((x+\log (25+x))^2\right )\right )\right )}-\frac {2 x^2}{(5+x) (x+\log (25+x)) \log \left ((x+\log (25+x))^2\right ) \left (e^x+x \log \left (\log \left ((x+\log (25+x))^2\right )\right )\right )}-\frac {25 x \log \left (\log \left ((x+\log (25+x))^2\right )\right )}{(5+x) (x+\log (25+x)) \left (e^x+x \log \left (\log \left ((x+\log (25+x))^2\right )\right )\right )}+\frac {24 x^2 \log \left (\log \left ((x+\log (25+x))^2\right )\right )}{(5+x) (x+\log (25+x)) \left (e^x+x \log \left (\log \left ((x+\log (25+x))^2\right )\right )\right )}+\frac {x^3 \log \left (\log \left ((x+\log (25+x))^2\right )\right )}{(5+x) (x+\log (25+x)) \left (e^x+x \log \left (\log \left ((x+\log (25+x))^2\right )\right )\right )}-\frac {25 \log (25+x) \log \left (\log \left ((x+\log (25+x))^2\right )\right )}{(5+x) (x+\log (25+x)) \left (e^x+x \log \left (\log \left ((x+\log (25+x))^2\right )\right )\right )}+\frac {24 x \log (25+x) \log \left (\log \left ((x+\log (25+x))^2\right )\right )}{(5+x) (x+\log (25+x)) \left (e^x+x \log \left (\log \left ((x+\log (25+x))^2\right )\right )\right )}+\frac {x^2 \log (25+x) \log \left (\log \left ((x+\log (25+x))^2\right )\right )}{(5+x) (x+\log (25+x)) \left (e^x+x \log \left (\log \left ((x+\log (25+x))^2\right )\right )\right )}\right ) \, dx+\frac {1}{20} \int \left (-\frac {52 x}{(25+x) (x+\log (25+x)) \log \left ((x+\log (25+x))^2\right ) \left (e^x+x \log \left (\log \left ((x+\log (25+x))^2\right )\right )\right )}-\frac {2 x^2}{(25+x) (x+\log (25+x)) \log \left ((x+\log (25+x))^2\right ) \left (e^x+x \log \left (\log \left ((x+\log (25+x))^2\right )\right )\right )}-\frac {25 x \log \left (\log \left ((x+\log (25+x))^2\right )\right )}{(25+x) (x+\log (25+x)) \left (e^x+x \log \left (\log \left ((x+\log (25+x))^2\right )\right )\right )}+\frac {24 x^2 \log \left (\log \left ((x+\log (25+x))^2\right )\right )}{(25+x) (x+\log (25+x)) \left (e^x+x \log \left (\log \left ((x+\log (25+x))^2\right )\right )\right )}+\frac {x^3 \log \left (\log \left ((x+\log (25+x))^2\right )\right )}{(25+x) (x+\log (25+x)) \left (e^x+x \log \left (\log \left ((x+\log (25+x))^2\right )\right )\right )}-\frac {25 \log (25+x) \log \left (\log \left ((x+\log (25+x))^2\right )\right )}{(25+x) (x+\log (25+x)) \left (e^x+x \log \left (\log \left ((x+\log (25+x))^2\right )\right )\right )}+\frac {24 x \log (25+x) \log \left (\log \left ((x+\log (25+x))^2\right )\right )}{(25+x) (x+\log (25+x)) \left (e^x+x \log \left (\log \left ((x+\log (25+x))^2\right )\right )\right )}+\frac {x^2 \log (25+x) \log \left (\log \left ((x+\log (25+x))^2\right )\right )}{(25+x) (x+\log (25+x)) \left (e^x+x \log \left (\log \left ((x+\log (25+x))^2\right )\right )\right )}\right ) \, dx-\int \frac {\log \left (e^x+x \log \left (\log \left ((x+\log (25+x))^2\right )\right )\right )}{(5+x)^2} \, dx \\ & = \text {Too large to display} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.23 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.00 \[ \int \frac {260 x+62 x^2+2 x^3+\left (e^x \left (125 x+30 x^2+x^3\right )+e^x \left (125+30 x+x^2\right ) \log (25+x)\right ) \log \left (x^2+2 x \log (25+x)+\log ^2(25+x)\right )+\left (125 x+30 x^2+x^3+\left (125+30 x+x^2\right ) \log (25+x)\right ) \log \left (x^2+2 x \log (25+x)+\log ^2(25+x)\right ) \log \left (\log \left (x^2+2 x \log (25+x)+\log ^2(25+x)\right )\right )+\left (\left (e^x \left (-25 x-x^2\right )+e^x (-25-x) \log (25+x)\right ) \log \left (x^2+2 x \log (25+x)+\log ^2(25+x)\right )+\left (-25 x^2-x^3+\left (-25 x-x^2\right ) \log (25+x)\right ) \log \left (x^2+2 x \log (25+x)+\log ^2(25+x)\right ) \log \left (\log \left (x^2+2 x \log (25+x)+\log ^2(25+x)\right )\right )\right ) \log \left (e^x+x \log \left (\log \left (x^2+2 x \log (25+x)+\log ^2(25+x)\right )\right )\right )}{\left (e^x \left (625 x+275 x^2+35 x^3+x^4\right )+e^x \left (625+275 x+35 x^2+x^3\right ) \log (25+x)\right ) \log \left (x^2+2 x \log (25+x)+\log ^2(25+x)\right )+\left (625 x^2+275 x^3+35 x^4+x^5+\left (625 x+275 x^2+35 x^3+x^4\right ) \log (25+x)\right ) \log \left (x^2+2 x \log (25+x)+\log ^2(25+x)\right ) \log \left (\log \left (x^2+2 x \log (25+x)+\log ^2(25+x)\right )\right )} \, dx=\frac {\log \left (e^x+x \log \left (\log \left ((x+\log (25+x))^2\right )\right )\right )}{5+x} \]

[In]

Integrate[(260*x + 62*x^2 + 2*x^3 + (E^x*(125*x + 30*x^2 + x^3) + E^x*(125 + 30*x + x^2)*Log[25 + x])*Log[x^2
+ 2*x*Log[25 + x] + Log[25 + x]^2] + (125*x + 30*x^2 + x^3 + (125 + 30*x + x^2)*Log[25 + x])*Log[x^2 + 2*x*Log
[25 + x] + Log[25 + x]^2]*Log[Log[x^2 + 2*x*Log[25 + x] + Log[25 + x]^2]] + ((E^x*(-25*x - x^2) + E^x*(-25 - x
)*Log[25 + x])*Log[x^2 + 2*x*Log[25 + x] + Log[25 + x]^2] + (-25*x^2 - x^3 + (-25*x - x^2)*Log[25 + x])*Log[x^
2 + 2*x*Log[25 + x] + Log[25 + x]^2]*Log[Log[x^2 + 2*x*Log[25 + x] + Log[25 + x]^2]])*Log[E^x + x*Log[Log[x^2
+ 2*x*Log[25 + x] + Log[25 + x]^2]]])/((E^x*(625*x + 275*x^2 + 35*x^3 + x^4) + E^x*(625 + 275*x + 35*x^2 + x^3
)*Log[25 + x])*Log[x^2 + 2*x*Log[25 + x] + Log[25 + x]^2] + (625*x^2 + 275*x^3 + 35*x^4 + x^5 + (625*x + 275*x
^2 + 35*x^3 + x^4)*Log[25 + x])*Log[x^2 + 2*x*Log[25 + x] + Log[25 + x]^2]*Log[Log[x^2 + 2*x*Log[25 + x] + Log
[25 + x]^2]]),x]

[Out]

Log[E^x + x*Log[Log[(x + Log[25 + x])^2]]]/(5 + x)

Maple [C] (warning: unable to verify)

Result contains higher order function than in optimal. Order 9 vs. order 3.

Time = 0.09 (sec) , antiderivative size = 67, normalized size of antiderivative = 2.91

\[\frac {\ln \left (x \ln \left (2 \ln \left (\ln \left (x +25\right )+x \right )-\frac {i \pi \,\operatorname {csgn}\left (i \left (\ln \left (x +25\right )+x \right )^{2}\right ) {\left (-\operatorname {csgn}\left (i \left (\ln \left (x +25\right )+x \right )^{2}\right )+\operatorname {csgn}\left (i \left (\ln \left (x +25\right )+x \right )\right )\right )}^{2}}{2}\right )+{\mathrm e}^{x}\right )}{5+x}\]

[In]

int(((((-x^2-25*x)*ln(x+25)-x^3-25*x^2)*ln(ln(x+25)^2+2*x*ln(x+25)+x^2)*ln(ln(ln(x+25)^2+2*x*ln(x+25)+x^2))+((
-x-25)*exp(x)*ln(x+25)+(-x^2-25*x)*exp(x))*ln(ln(x+25)^2+2*x*ln(x+25)+x^2))*ln(x*ln(ln(ln(x+25)^2+2*x*ln(x+25)
+x^2))+exp(x))+((x^2+30*x+125)*ln(x+25)+x^3+30*x^2+125*x)*ln(ln(x+25)^2+2*x*ln(x+25)+x^2)*ln(ln(ln(x+25)^2+2*x
*ln(x+25)+x^2))+((x^2+30*x+125)*exp(x)*ln(x+25)+(x^3+30*x^2+125*x)*exp(x))*ln(ln(x+25)^2+2*x*ln(x+25)+x^2)+2*x
^3+62*x^2+260*x)/(((x^4+35*x^3+275*x^2+625*x)*ln(x+25)+x^5+35*x^4+275*x^3+625*x^2)*ln(ln(x+25)^2+2*x*ln(x+25)+
x^2)*ln(ln(ln(x+25)^2+2*x*ln(x+25)+x^2))+((x^3+35*x^2+275*x+625)*exp(x)*ln(x+25)+(x^4+35*x^3+275*x^2+625*x)*ex
p(x))*ln(ln(x+25)^2+2*x*ln(x+25)+x^2)),x)

[Out]

1/(5+x)*ln(x*ln(2*ln(ln(x+25)+x)-1/2*I*Pi*csgn(I*(ln(x+25)+x)^2)*(-csgn(I*(ln(x+25)+x)^2)+csgn(I*(ln(x+25)+x))
)^2)+exp(x))

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.35 \[ \int \frac {260 x+62 x^2+2 x^3+\left (e^x \left (125 x+30 x^2+x^3\right )+e^x \left (125+30 x+x^2\right ) \log (25+x)\right ) \log \left (x^2+2 x \log (25+x)+\log ^2(25+x)\right )+\left (125 x+30 x^2+x^3+\left (125+30 x+x^2\right ) \log (25+x)\right ) \log \left (x^2+2 x \log (25+x)+\log ^2(25+x)\right ) \log \left (\log \left (x^2+2 x \log (25+x)+\log ^2(25+x)\right )\right )+\left (\left (e^x \left (-25 x-x^2\right )+e^x (-25-x) \log (25+x)\right ) \log \left (x^2+2 x \log (25+x)+\log ^2(25+x)\right )+\left (-25 x^2-x^3+\left (-25 x-x^2\right ) \log (25+x)\right ) \log \left (x^2+2 x \log (25+x)+\log ^2(25+x)\right ) \log \left (\log \left (x^2+2 x \log (25+x)+\log ^2(25+x)\right )\right )\right ) \log \left (e^x+x \log \left (\log \left (x^2+2 x \log (25+x)+\log ^2(25+x)\right )\right )\right )}{\left (e^x \left (625 x+275 x^2+35 x^3+x^4\right )+e^x \left (625+275 x+35 x^2+x^3\right ) \log (25+x)\right ) \log \left (x^2+2 x \log (25+x)+\log ^2(25+x)\right )+\left (625 x^2+275 x^3+35 x^4+x^5+\left (625 x+275 x^2+35 x^3+x^4\right ) \log (25+x)\right ) \log \left (x^2+2 x \log (25+x)+\log ^2(25+x)\right ) \log \left (\log \left (x^2+2 x \log (25+x)+\log ^2(25+x)\right )\right )} \, dx=\frac {\log \left (x \log \left (\log \left (x^{2} + 2 \, x \log \left (x + 25\right ) + \log \left (x + 25\right )^{2}\right )\right ) + e^{x}\right )}{x + 5} \]

[In]

integrate(((((-x^2-25*x)*log(x+25)-x^3-25*x^2)*log(log(x+25)^2+2*x*log(x+25)+x^2)*log(log(log(x+25)^2+2*x*log(
x+25)+x^2))+((-x-25)*exp(x)*log(x+25)+(-x^2-25*x)*exp(x))*log(log(x+25)^2+2*x*log(x+25)+x^2))*log(x*log(log(lo
g(x+25)^2+2*x*log(x+25)+x^2))+exp(x))+((x^2+30*x+125)*log(x+25)+x^3+30*x^2+125*x)*log(log(x+25)^2+2*x*log(x+25
)+x^2)*log(log(log(x+25)^2+2*x*log(x+25)+x^2))+((x^2+30*x+125)*exp(x)*log(x+25)+(x^3+30*x^2+125*x)*exp(x))*log
(log(x+25)^2+2*x*log(x+25)+x^2)+2*x^3+62*x^2+260*x)/(((x^4+35*x^3+275*x^2+625*x)*log(x+25)+x^5+35*x^4+275*x^3+
625*x^2)*log(log(x+25)^2+2*x*log(x+25)+x^2)*log(log(log(x+25)^2+2*x*log(x+25)+x^2))+((x^3+35*x^2+275*x+625)*ex
p(x)*log(x+25)+(x^4+35*x^3+275*x^2+625*x)*exp(x))*log(log(x+25)^2+2*x*log(x+25)+x^2)),x, algorithm="fricas")

[Out]

log(x*log(log(x^2 + 2*x*log(x + 25) + log(x + 25)^2)) + e^x)/(x + 5)

Sympy [F(-1)]

Timed out. \[ \int \frac {260 x+62 x^2+2 x^3+\left (e^x \left (125 x+30 x^2+x^3\right )+e^x \left (125+30 x+x^2\right ) \log (25+x)\right ) \log \left (x^2+2 x \log (25+x)+\log ^2(25+x)\right )+\left (125 x+30 x^2+x^3+\left (125+30 x+x^2\right ) \log (25+x)\right ) \log \left (x^2+2 x \log (25+x)+\log ^2(25+x)\right ) \log \left (\log \left (x^2+2 x \log (25+x)+\log ^2(25+x)\right )\right )+\left (\left (e^x \left (-25 x-x^2\right )+e^x (-25-x) \log (25+x)\right ) \log \left (x^2+2 x \log (25+x)+\log ^2(25+x)\right )+\left (-25 x^2-x^3+\left (-25 x-x^2\right ) \log (25+x)\right ) \log \left (x^2+2 x \log (25+x)+\log ^2(25+x)\right ) \log \left (\log \left (x^2+2 x \log (25+x)+\log ^2(25+x)\right )\right )\right ) \log \left (e^x+x \log \left (\log \left (x^2+2 x \log (25+x)+\log ^2(25+x)\right )\right )\right )}{\left (e^x \left (625 x+275 x^2+35 x^3+x^4\right )+e^x \left (625+275 x+35 x^2+x^3\right ) \log (25+x)\right ) \log \left (x^2+2 x \log (25+x)+\log ^2(25+x)\right )+\left (625 x^2+275 x^3+35 x^4+x^5+\left (625 x+275 x^2+35 x^3+x^4\right ) \log (25+x)\right ) \log \left (x^2+2 x \log (25+x)+\log ^2(25+x)\right ) \log \left (\log \left (x^2+2 x \log (25+x)+\log ^2(25+x)\right )\right )} \, dx=\text {Timed out} \]

[In]

integrate(((((-x**2-25*x)*ln(x+25)-x**3-25*x**2)*ln(ln(x+25)**2+2*x*ln(x+25)+x**2)*ln(ln(ln(x+25)**2+2*x*ln(x+
25)+x**2))+((-x-25)*exp(x)*ln(x+25)+(-x**2-25*x)*exp(x))*ln(ln(x+25)**2+2*x*ln(x+25)+x**2))*ln(x*ln(ln(ln(x+25
)**2+2*x*ln(x+25)+x**2))+exp(x))+((x**2+30*x+125)*ln(x+25)+x**3+30*x**2+125*x)*ln(ln(x+25)**2+2*x*ln(x+25)+x**
2)*ln(ln(ln(x+25)**2+2*x*ln(x+25)+x**2))+((x**2+30*x+125)*exp(x)*ln(x+25)+(x**3+30*x**2+125*x)*exp(x))*ln(ln(x
+25)**2+2*x*ln(x+25)+x**2)+2*x**3+62*x**2+260*x)/(((x**4+35*x**3+275*x**2+625*x)*ln(x+25)+x**5+35*x**4+275*x**
3+625*x**2)*ln(ln(x+25)**2+2*x*ln(x+25)+x**2)*ln(ln(ln(x+25)**2+2*x*ln(x+25)+x**2))+((x**3+35*x**2+275*x+625)*
exp(x)*ln(x+25)+(x**4+35*x**3+275*x**2+625*x)*exp(x))*ln(ln(x+25)**2+2*x*ln(x+25)+x**2)),x)

[Out]

Timed out

Maxima [A] (verification not implemented)

none

Time = 0.43 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.00 \[ \int \frac {260 x+62 x^2+2 x^3+\left (e^x \left (125 x+30 x^2+x^3\right )+e^x \left (125+30 x+x^2\right ) \log (25+x)\right ) \log \left (x^2+2 x \log (25+x)+\log ^2(25+x)\right )+\left (125 x+30 x^2+x^3+\left (125+30 x+x^2\right ) \log (25+x)\right ) \log \left (x^2+2 x \log (25+x)+\log ^2(25+x)\right ) \log \left (\log \left (x^2+2 x \log (25+x)+\log ^2(25+x)\right )\right )+\left (\left (e^x \left (-25 x-x^2\right )+e^x (-25-x) \log (25+x)\right ) \log \left (x^2+2 x \log (25+x)+\log ^2(25+x)\right )+\left (-25 x^2-x^3+\left (-25 x-x^2\right ) \log (25+x)\right ) \log \left (x^2+2 x \log (25+x)+\log ^2(25+x)\right ) \log \left (\log \left (x^2+2 x \log (25+x)+\log ^2(25+x)\right )\right )\right ) \log \left (e^x+x \log \left (\log \left (x^2+2 x \log (25+x)+\log ^2(25+x)\right )\right )\right )}{\left (e^x \left (625 x+275 x^2+35 x^3+x^4\right )+e^x \left (625+275 x+35 x^2+x^3\right ) \log (25+x)\right ) \log \left (x^2+2 x \log (25+x)+\log ^2(25+x)\right )+\left (625 x^2+275 x^3+35 x^4+x^5+\left (625 x+275 x^2+35 x^3+x^4\right ) \log (25+x)\right ) \log \left (x^2+2 x \log (25+x)+\log ^2(25+x)\right ) \log \left (\log \left (x^2+2 x \log (25+x)+\log ^2(25+x)\right )\right )} \, dx=\frac {\log \left (x {\left (\log \left (2\right ) + \log \left (\log \left (x + \log \left (x + 25\right )\right )\right )\right )} + e^{x}\right )}{x + 5} \]

[In]

integrate(((((-x^2-25*x)*log(x+25)-x^3-25*x^2)*log(log(x+25)^2+2*x*log(x+25)+x^2)*log(log(log(x+25)^2+2*x*log(
x+25)+x^2))+((-x-25)*exp(x)*log(x+25)+(-x^2-25*x)*exp(x))*log(log(x+25)^2+2*x*log(x+25)+x^2))*log(x*log(log(lo
g(x+25)^2+2*x*log(x+25)+x^2))+exp(x))+((x^2+30*x+125)*log(x+25)+x^3+30*x^2+125*x)*log(log(x+25)^2+2*x*log(x+25
)+x^2)*log(log(log(x+25)^2+2*x*log(x+25)+x^2))+((x^2+30*x+125)*exp(x)*log(x+25)+(x^3+30*x^2+125*x)*exp(x))*log
(log(x+25)^2+2*x*log(x+25)+x^2)+2*x^3+62*x^2+260*x)/(((x^4+35*x^3+275*x^2+625*x)*log(x+25)+x^5+35*x^4+275*x^3+
625*x^2)*log(log(x+25)^2+2*x*log(x+25)+x^2)*log(log(log(x+25)^2+2*x*log(x+25)+x^2))+((x^3+35*x^2+275*x+625)*ex
p(x)*log(x+25)+(x^4+35*x^3+275*x^2+625*x)*exp(x))*log(log(x+25)^2+2*x*log(x+25)+x^2)),x, algorithm="maxima")

[Out]

log(x*(log(2) + log(log(x + log(x + 25)))) + e^x)/(x + 5)

Giac [A] (verification not implemented)

none

Time = 1.58 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.35 \[ \int \frac {260 x+62 x^2+2 x^3+\left (e^x \left (125 x+30 x^2+x^3\right )+e^x \left (125+30 x+x^2\right ) \log (25+x)\right ) \log \left (x^2+2 x \log (25+x)+\log ^2(25+x)\right )+\left (125 x+30 x^2+x^3+\left (125+30 x+x^2\right ) \log (25+x)\right ) \log \left (x^2+2 x \log (25+x)+\log ^2(25+x)\right ) \log \left (\log \left (x^2+2 x \log (25+x)+\log ^2(25+x)\right )\right )+\left (\left (e^x \left (-25 x-x^2\right )+e^x (-25-x) \log (25+x)\right ) \log \left (x^2+2 x \log (25+x)+\log ^2(25+x)\right )+\left (-25 x^2-x^3+\left (-25 x-x^2\right ) \log (25+x)\right ) \log \left (x^2+2 x \log (25+x)+\log ^2(25+x)\right ) \log \left (\log \left (x^2+2 x \log (25+x)+\log ^2(25+x)\right )\right )\right ) \log \left (e^x+x \log \left (\log \left (x^2+2 x \log (25+x)+\log ^2(25+x)\right )\right )\right )}{\left (e^x \left (625 x+275 x^2+35 x^3+x^4\right )+e^x \left (625+275 x+35 x^2+x^3\right ) \log (25+x)\right ) \log \left (x^2+2 x \log (25+x)+\log ^2(25+x)\right )+\left (625 x^2+275 x^3+35 x^4+x^5+\left (625 x+275 x^2+35 x^3+x^4\right ) \log (25+x)\right ) \log \left (x^2+2 x \log (25+x)+\log ^2(25+x)\right ) \log \left (\log \left (x^2+2 x \log (25+x)+\log ^2(25+x)\right )\right )} \, dx=\frac {\log \left (x \log \left (\log \left (x^{2} + 2 \, x \log \left (x + 25\right ) + \log \left (x + 25\right )^{2}\right )\right ) + e^{x}\right )}{x + 5} \]

[In]

integrate(((((-x^2-25*x)*log(x+25)-x^3-25*x^2)*log(log(x+25)^2+2*x*log(x+25)+x^2)*log(log(log(x+25)^2+2*x*log(
x+25)+x^2))+((-x-25)*exp(x)*log(x+25)+(-x^2-25*x)*exp(x))*log(log(x+25)^2+2*x*log(x+25)+x^2))*log(x*log(log(lo
g(x+25)^2+2*x*log(x+25)+x^2))+exp(x))+((x^2+30*x+125)*log(x+25)+x^3+30*x^2+125*x)*log(log(x+25)^2+2*x*log(x+25
)+x^2)*log(log(log(x+25)^2+2*x*log(x+25)+x^2))+((x^2+30*x+125)*exp(x)*log(x+25)+(x^3+30*x^2+125*x)*exp(x))*log
(log(x+25)^2+2*x*log(x+25)+x^2)+2*x^3+62*x^2+260*x)/(((x^4+35*x^3+275*x^2+625*x)*log(x+25)+x^5+35*x^4+275*x^3+
625*x^2)*log(log(x+25)^2+2*x*log(x+25)+x^2)*log(log(log(x+25)^2+2*x*log(x+25)+x^2))+((x^3+35*x^2+275*x+625)*ex
p(x)*log(x+25)+(x^4+35*x^3+275*x^2+625*x)*exp(x))*log(log(x+25)^2+2*x*log(x+25)+x^2)),x, algorithm="giac")

[Out]

log(x*log(log(x^2 + 2*x*log(x + 25) + log(x + 25)^2)) + e^x)/(x + 5)

Mupad [B] (verification not implemented)

Time = 10.78 (sec) , antiderivative size = 54, normalized size of antiderivative = 2.35 \[ \int \frac {260 x+62 x^2+2 x^3+\left (e^x \left (125 x+30 x^2+x^3\right )+e^x \left (125+30 x+x^2\right ) \log (25+x)\right ) \log \left (x^2+2 x \log (25+x)+\log ^2(25+x)\right )+\left (125 x+30 x^2+x^3+\left (125+30 x+x^2\right ) \log (25+x)\right ) \log \left (x^2+2 x \log (25+x)+\log ^2(25+x)\right ) \log \left (\log \left (x^2+2 x \log (25+x)+\log ^2(25+x)\right )\right )+\left (\left (e^x \left (-25 x-x^2\right )+e^x (-25-x) \log (25+x)\right ) \log \left (x^2+2 x \log (25+x)+\log ^2(25+x)\right )+\left (-25 x^2-x^3+\left (-25 x-x^2\right ) \log (25+x)\right ) \log \left (x^2+2 x \log (25+x)+\log ^2(25+x)\right ) \log \left (\log \left (x^2+2 x \log (25+x)+\log ^2(25+x)\right )\right )\right ) \log \left (e^x+x \log \left (\log \left (x^2+2 x \log (25+x)+\log ^2(25+x)\right )\right )\right )}{\left (e^x \left (625 x+275 x^2+35 x^3+x^4\right )+e^x \left (625+275 x+35 x^2+x^3\right ) \log (25+x)\right ) \log \left (x^2+2 x \log (25+x)+\log ^2(25+x)\right )+\left (625 x^2+275 x^3+35 x^4+x^5+\left (625 x+275 x^2+35 x^3+x^4\right ) \log (25+x)\right ) \log \left (x^2+2 x \log (25+x)+\log ^2(25+x)\right ) \log \left (\log \left (x^2+2 x \log (25+x)+\log ^2(25+x)\right )\right )} \, dx=\frac {\ln \left ({\mathrm {e}}^x+x\,\ln \left (\ln \left (x^2+2\,x\,\ln \left (x+25\right )+{\ln \left (x+25\right )}^2\right )\right )\right )\,\left (x^2+25\,x\right )\,\left (x^2+30\,x+125\right )}{x\,{\left (x+5\right )}^2\,{\left (x+25\right )}^2} \]

[In]

int((260*x + log(2*x*log(x + 25) + log(x + 25)^2 + x^2)*(exp(x)*(125*x + 30*x^2 + x^3) + log(x + 25)*exp(x)*(3
0*x + x^2 + 125)) - log(exp(x) + x*log(log(2*x*log(x + 25) + log(x + 25)^2 + x^2)))*(log(2*x*log(x + 25) + log
(x + 25)^2 + x^2)*(exp(x)*(25*x + x^2) + log(x + 25)*exp(x)*(x + 25)) + log(2*x*log(x + 25) + log(x + 25)^2 +
x^2)*log(log(2*x*log(x + 25) + log(x + 25)^2 + x^2))*(log(x + 25)*(25*x + x^2) + 25*x^2 + x^3)) + 62*x^2 + 2*x
^3 + log(2*x*log(x + 25) + log(x + 25)^2 + x^2)*log(log(2*x*log(x + 25) + log(x + 25)^2 + x^2))*(125*x + log(x
+ 25)*(30*x + x^2 + 125) + 30*x^2 + x^3))/(log(2*x*log(x + 25) + log(x + 25)^2 + x^2)*(exp(x)*(625*x + 275*x^
2 + 35*x^3 + x^4) + log(x + 25)*exp(x)*(275*x + 35*x^2 + x^3 + 625)) + log(2*x*log(x + 25) + log(x + 25)^2 + x
^2)*log(log(2*x*log(x + 25) + log(x + 25)^2 + x^2))*(log(x + 25)*(625*x + 275*x^2 + 35*x^3 + x^4) + 625*x^2 +
275*x^3 + 35*x^4 + x^5)),x)

[Out]

(log(exp(x) + x*log(log(2*x*log(x + 25) + log(x + 25)^2 + x^2)))*(25*x + x^2)*(30*x + x^2 + 125))/(x*(x + 5)^2
*(x + 25)^2)